#Curious about why we can choose epsilons when determining if there is a limit in a function?

11 messages · Page 1 of 1 (latest)

broken mesa
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Hey guys, sorry its maybe a dumb question. I've been reviewing some limits on Khan academy and to understand proving limits he presented:

"For any range around L you give me (Epsilon), i have to find you a range of Xs around C (delta) that will make f(x) within the epsilon of L"

So if epsilon is 0.1, then i have to find a range of f(x) that puts you between L-0.1 and L+0.1

But my question is, that if 0.1 is true, and every smaller epsilon must be true, then why even bother choosing an epsilon? Why not just always default to "epsilon is as small as humanly possible"?

My reasoning is:

if you choose an infinitely small epsilon, and there is a limit for it, then it necessitates that every larger epsilon will also be true. If that's the case, why even bother checking epsilon = 0.1, epsilon = 0.001, or any of those? Since we can use "epsilon is as small as possible" as a blanket statement to cover all possible epsilons?

Because to me with that explanation epsilons sound kind of like p-values, where i can just choose a "good enough" value based on context and say well within this range the limit is true.

I'm sorry if my wording or assumptions are wrong, I hope my question makes sense😅 , thank you in advance guys 🙏

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merry lotus
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there is no "smallest number" is the thing

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but the way you actually prove that a limit exists is to find delta as a function of epsilon, so that epsilon can be made as small as you want it without settling on a specific value

broken mesa
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Ahh i see, thank you! so in theory its proveable with E > 0, but for any given graph in real life its impossible to truly prove a limit then? For that specific graph? So we just take a really small epsilon and call it good enough?

merry lotus
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no?

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the definition of a limit simply says that "given any epsilon, no matter how small, a delta can be found"

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so there is no need to set "epsilon = smallest possible number" because it isn't a fixed number at all

broken mesa
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but in practical scenarios where we are splitting the graph into many small rectangles, the "width" of each rectangle is the delta right? So for the calculations, we'd need some kind of epsilon / delta or "smallness" to go off of right?

broken mesa
merry lotus
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for formal limit proofs you don't just "test values of epsilon", you find delta as a function of epsilon